2. Side Angle Side Theorem By Andrew Moser

3. Summary  If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.  If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.  If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.  If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

4. Examples

5. Web Links  http://www.mathwarehouse.com/trigonomet ry/area/side-angle-side-triangle.html http://www.mathwarehouse.com/trigonomet ry/area/side-angle-side-triangle.html  http://hotmath.com/hotmath_help/topics/SA S-postulate.html http://hotmath.com/hotmath_help/topics/SA S-postulate.html  http://www.jimloy.com/cindy/ass.htm http://www.jimloy.com/cindy/ass.htm  http://www.mathwarehouse.com/trigonomet ry/area/side-angle-side-triangle.html http://www.mathwarehouse.com/trigonomet ry/area/side-angle-side-triangle.html  http://hotmath.com/hotmath_help/topics/SA S-postulate.html http://hotmath.com/hotmath_help/topics/SA S-postulate.html  http://www.jimloy.com/cindy/ass.htm http://www.jimloy.com/cindy/ass.htm

6. Side Side Side Kyle Schroeder

7. Summary  You can only find SSS if the three sides in one triangle are congruent.  We learned this when using Solving Triangle Proofs  You can only find SSS if the three sides in one triangle are congruent.  We learned this when using Solving Triangle Proofs

8. Rules, Properties, & Formulas  The rule and property for SSS theorem is that you can only determine that you have reached SSS is that the triangle has to be congruent to the other triangle

10. Web Links  http://www.cut-the- knot.org/pythagoras/SSS.shtml http://www.cut-the- knot.org/pythagoras/SSS.shtml  http://www.tutorvista.com/topic/proof-of- sss-theorem http://www.tutorvista.com/topic/proof-of- sss-theorem  http://www.mathwarehouse.com/geometry/ congruent_triangles/side-side-side- postulate.php http://www.mathwarehouse.com/geometry/ congruent_triangles/side-side-side- postulate.php  http://www.cut-the- knot.org/pythagoras/SSS.shtml http://www.cut-the- knot.org/pythagoras/SSS.shtml  http://www.tutorvista.com/topic/proof-of- sss-theorem http://www.tutorvista.com/topic/proof-of- sss-theorem  http://www.mathwarehouse.com/geometry/ congruent_triangles/side-side-side- postulate.php http://www.mathwarehouse.com/geometry/ congruent_triangles/side-side-side- postulate.php

11. Proofs Involving CPCTC by, Nick Karach Summary: -CPCTC stands for: “ C orresponding P arts of C orresponding T riangles are C ongruent”  This means that once you prove two triangle congruent, you know that corresponding sides and angles are congruent. Summary: -CPCTC stands for: “ C orresponding P arts of C orresponding T riangles are C ongruent”  This means that once you prove two triangle congruent, you know that corresponding sides and angles are congruent.

12. Rules, Properties & Formulas  First of all you must prove the Triangles congruent through a postulate such as ASA, SAS, AAS or HL.  Second, once you state the two triangles are congruent, you can state a two sides are congruent. Ex.  First of all you must prove the Triangles congruent through a postulate such as ASA, SAS, AAS or HL.  Second, once you state the two triangles are congruent, you can state a two sides are congruent. Ex.

13. Examples Given :

14. Web Links  Main Concept and Some Examples Main Concept and Some Examples  CPCTC WikiPedia CPCTC WikiPedia  Examples Examples  Main Concept and Some Examples Main Concept and Some Examples  CPCTC WikiPedia CPCTC WikiPedia  Examples Examples

15. Equilateral Triangle By Jake Morra

16. Equilateral Triangles  A equilateral triangle is a triangle where all the sides are equal in length.  All angles opposite though sides are congruent  A equilateral triangle is a triangle where all the sides are equal in length.  All angles opposite though sides are congruent

17. Finding The Height To find the height add an altitude from vertexes to opposite segment If Segment AB, BC, and CA are all 10 then Segment BP and PC are 5 If the added segment is a altitude. angle BPA and APC are 90 degrees Now that you know all of this can solve the height by the Pythagorean Theorem a + 5 = 10

18. Other Websites To Help You  http://mathcentral.uregina.ca/QQ/database/ QQ.09.02/rosa2.html http://mathcentral.uregina.ca/QQ/database/ QQ.09.02/rosa2.html  www.calcenstein.com/calc/1111_help.php www.calcenstein.com/calc/1111_help.php  www.ehow.com › Education › Math Education › Triangles  http://mathcentral.uregina.ca/QQ/database/ QQ.09.02/rosa2.html http://mathcentral.uregina.ca/QQ/database/ QQ.09.02/rosa2.html  www.calcenstein.com/calc/1111_help.php www.calcenstein.com/calc/1111_help.php  www.ehow.com › Education › Math Education › Triangles

19. Angle Bisector and Incenter -What is an angle bisector and an incenter? -Example problems -Web links -What is an angle bisector and an incenter? -Example problems -Web links

20. What is an angle bisector and an incenter? An angle bisector is a segment that divided an angle in half. When the three angle bisectors intersect they create a point of concurrency which is called the incenter

21. Ex: 1- Both little angles will be the same measure

22. Ex: 2 Find x Equation: 13x-1= 2(6x+4) 13x-1= 12x+8 -12x 12x x-1= 8 +1 +1 X= 9 Equation: 13x-1= 2(6x+4) 13x-1= 12x+8 -12x 12x x-1= 8 +1 +1 X= 9

23. Ex:3 Incenter is ALWAYS in the middle Acute Right Obtuse

24. Helpful Links  http://www.cliffsnotes.com/study_guide/Altitudes-Medians-and-Angle- Bisectors.topicArticleId-18851,articleId-18787.html http://www.cliffsnotes.com/study_guide/Altitudes-Medians-and-Angle- Bisectors.topicArticleId-18851,articleId-18787.html  http://jwilson.coe.uga.edu/emt725/Prob.2.35.1/Problem.2.35.1.html http://jwilson.coe.uga.edu/emt725/Prob.2.35.1/Problem.2.35.1.html  http://mathworld.wolfram.com/AngleBisector.html http://mathworld.wolfram.com/AngleBisector.html  http://www.cliffsnotes.com/study_guide/Altitudes-Medians-and-Angle- Bisectors.topicArticleId-18851,articleId-18787.html http://www.cliffsnotes.com/study_guide/Altitudes-Medians-and-Angle- Bisectors.topicArticleId-18851,articleId-18787.html  http://jwilson.coe.uga.edu/emt725/Prob.2.35.1/Problem.2.35.1.html http://jwilson.coe.uga.edu/emt725/Prob.2.35.1/Problem.2.35.1.html  http://mathworld.wolfram.com/AngleBisector.html http://mathworld.wolfram.com/AngleBisector.html

25. Angle Side Angle Theorem By: Daulton Moro

26. AAS Theorem Summary:  The AAS theorem is one of the theorems that is used to prove triangles congruent.  The AAS theorem is when two angles and one non-included side are congruent.  The AAS theorem is one of the theorems that is used to prove triangles congruent.  The AAS theorem is when two angles and one non-included side are congruent.

27. Sample Problems  For the first picture you would mark lines BC and CE congruent and angles A and D would be congruent. After mark the vertical angles congruent the you have congruence by AAS.  The second picture shows AAS because there are two angles that are congruent and one side that is non-included.  The third picture is self explanatory and is proven by using AAS.  For the first picture you would mark lines BC and CE congruent and angles A and D would be congruent. After mark the vertical angles congruent the you have congruence by AAS.  The second picture shows AAS because there are two angles that are congruent and one side that is non-included.  The third picture is self explanatory and is proven by using AAS.

28. Helpful Websites  www.mathwarehouse.com www.mathwarehouse.com  www.library.thinkquest.org www.library.thinkquest.org  www.phschool.com www.phschool.com ww ww.mathwarehouse.com ww ww.library.thinkquest.org ww ww.phschool.com

29. What exactly is an HL proof? By Dylan Sen  The hypotenuse leg theorem, or HL, is the congruence theorem used to prove only right triangles congruent.  Also The theorem states that any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg are congruent triangles..  The goal of today’s lesson is to prove right triangles congruent using the HL theorem  The hypotenuse leg theorem, or HL, is the congruence theorem used to prove only right triangles congruent.  Also The theorem states that any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg are congruent triangles..  The goal of today’s lesson is to prove right triangles congruent using the HL theorem

30. Rules and Formulas  As seen in the previous slide, if the hypotenuse and leg of one triangle are congruent to the hypotenuse and leg of the other, the triangles are congruent.  The most important formula to remember is:  As seen in the previous slide, if the hypotenuse and leg of one triangle are congruent to the hypotenuse and leg of the other, the triangles are congruent.  The most important formula to remember is:

31. Examples Given: Prove: StatementReason (leg)-Given (hypotenuse)- Given and-They both have a right angle. are right triangles - Through the HL theorem. Since the hypotenuse and the leg are congruent, that means the triangles are congruent Given: Prove: StatementReason (leg)-Given (hypotenuse)- Given and-They both have a right angle. are right triangles - Through the HL theorem. Since the hypotenuse and the leg are congruent, that means the triangles are congruent

32. Given- and Prove: (leg)Given (hypotenuse) Given and They have a right angle are right triangles Because the hypotenuse and corresponding leg are congruent, the triangles are congruent Given- and Prove: (leg)Given (hypotenuse) Given and They have a right angle are right triangles Because the hypotenuse and corresponding leg are congruent, the triangles are congruent

33. Given: and Prove StatementReason (leg)Given (hypotenuse) Given and They have a right angle are right triangles Because the hypotenuse and corresponding leg are congruent, the triangles are congruent Given: and Prove StatementReason (leg)Given (hypotenuse) Given and They have a right angle are right triangles Because the hypotenuse and corresponding leg are congruent, the triangles are congruent

34. Useful Websites to help you further understand HL:  http://delta.classwell.com/ebooks/navigateBook.clg?sectionType=unit&na vigation=1&prevNext=0&curSeq=235&curDispPage=239&xpqData=%2 Fcontent%5B%40id%3D%27mcd_ma_geo_lsn_0395937779_p236.xml% 27%5D - This is the textbook definition. It will show examples and a step by step method of figuring out how to use HL. http://delta.classwell.com/ebooks/navigateBook.clg?sectionType=unit&na vigation=1&prevNext=0&curSeq=235&curDispPage=239&xpqData=%2 Fcontent%5B%40id%3D%27mcd_ma_geo_lsn_0395937779_p236.xml% 27%5D  http://www.mathwarehouse.com/geometry/congruent_triangles/hypotenus e-leg-theorem.php - Much like the textbook, this website shows great examples and will help clarify anything you have trouble with. http://www.mathwarehouse.com/geometry/congruent_triangles/hypotenus e-leg-theorem.php  http://www.onlinemathlearning.com/hypotenuse-leg.html - this example shows more guided examples, which will further help you understand the HL Theorem http://www.onlinemathlearning.com/hypotenuse-leg.html  http://delta.classwell.com/ebooks/navigateBook.clg?sectionType=unit&na vigation=1&prevNext=0&curSeq=235&curDispPage=239&xpqData=%2 Fcontent%5B%40id%3D%27mcd_ma_geo_lsn_0395937779_p236.xml% 27%5D - This is the textbook definition. It will show examples and a step by step method of figuring out how to use HL. http://delta.classwell.com/ebooks/navigateBook.clg?sectionType=unit&na vigation=1&prevNext=0&curSeq=235&curDispPage=239&xpqData=%2 Fcontent%5B%40id%3D%27mcd_ma_geo_lsn_0395937779_p236.xml% 27%5D  http://www.mathwarehouse.com/geometry/congruent_triangles/hypotenus e-leg-theorem.php - Much like the textbook, this website shows great examples and will help clarify anything you have trouble with. http://www.mathwarehouse.com/geometry/congruent_triangles/hypotenus e-leg-theorem.php  http://www.onlinemathlearning.com/hypotenuse-leg.html - this example shows more guided examples, which will further help you understand the HL Theorem http://www.onlinemathlearning.com/hypotenuse-leg.html

35. Medians and Centroids Summary: A median is a segment that connects the vertex of a triangle to the midpoint of the opposite side. The point of concurrency (intersection) of the medians is called the centroid. Goals: The goals of this presentation are to: 1) Review Medians and Centroids 2) Review Sample Problems

36. Medians and Centroids  A median is a segment that connects the vertex of a triangle to the midpoint of the opposite side  The point of concurrency (intersection) of the medians is called the centroid  The distance from the vertex to the centroid is 2/3 of the total distance of the median  No matter what type of triangle (right, acute, obtuse), the centroid is ALWAYS inside the triangle  A median is a segment that connects the vertex of a triangle to the midpoint of the opposite side  The point of concurrency (intersection) of the medians is called the centroid  The distance from the vertex to the centroid is 2/3 of the total distance of the median  No matter what type of triangle (right, acute, obtuse), the centroid is ALWAYS inside the triangle

37. Sample Problems 1) Always, Sometimes, Never: The centroid ________________ lies within the triangle. 2) Find x: 3) Fill In The Blank: A triangle has ____________ medians. 1) Always, Sometimes, Never: The centroid ________________ lies within the triangle. 2) Find x: 3) Fill In The Blank: A triangle has ____________ medians.

38. Helpful Links  http://www.mathopenref.com/trianglemedians.html http://www.mathopenref.com/trianglemedians.html  http://mathworld.wolfram.com/TriangleMedian.html http://mathworld.wolfram.com/TriangleMedian.html  http://www.analyzemath.com/Geometry/MediansTriangle/ MediansTriangle.html http://www.analyzemath.com/Geometry/MediansTriangle/ MediansTriangle.html  http://www.cut-the-knot.org/triangle/medians.shtml http://www.cut-the-knot.org/triangle/medians.shtml  http://www.mathopenref.com/trianglemedians.html http://www.mathopenref.com/trianglemedians.html  http://mathworld.wolfram.com/TriangleMedian.html http://mathworld.wolfram.com/TriangleMedian.html  http://www.analyzemath.com/Geometry/MediansTriangle/ MediansTriangle.html http://www.analyzemath.com/Geometry/MediansTriangle/ MediansTriangle.html  http://www.cut-the-knot.org/triangle/medians.shtml http://www.cut-the-knot.org/triangle/medians.shtml